Open Access

Unitarily invariant norm inequalities involving Heron and Heinz means

Journal of Inequalities and Applications20142014:288

https://doi.org/10.1186/1029-242X-2014-288

Received: 16 April 2014

Accepted: 4 July 2014

Published: 18 August 2014

Abstract

In this paper, we present some new inequalities for unitarily invariant norms involving Heron and Heinz means for matrices, which generalize the result of Theorem 2.1 (Fu and He in J. Math. Inequal. 7(4):727-737, 2013) and refine the inequality of Theorem 6 (Zhan in SIAM J. Matrix Anal. Appl. 20: 466-470, 1998). Our results are a refinement and a generalization of some existing inequalities.

MSC:47A30, 15A60.

Keywords

unitarily invariant normpositive definite matricesconvex functionHeron meansHeinz means

1 Introduction

Throughout, let M m , n be the space of m × n complex matrices and M n = M n , n .

A norm is called unitarily invariant norm if U A V = A for all A M n and for all unitary matrices U , V M n . Two classes of unitarily invariant norms are especially important. The first is the class of the Ky Fan k-norm ( k ) , defined as
A ( k ) = j = 1 k s j ( A ) , k = 1 , , n ,
where s i ( A ) ( i = 1 , , n ) are the singular values of A with s 1 ( A ) s n ( A ) , which are the eigenvalues of the positive semidefinite matrix | A | = ( A A ) 1 2 , arranged in decreasing order and repeated according to multiplicity. The second is the class of the Schatten p-norm ( p ) , defined as
A p = ( j = 1 n s j p ( A ) ) 1 p = ( tr | A | p ) 1 p , 1 p < .
For two nonnegative real numbers a and b, the Heinz mean and Heron mean in the parameter v, 0 v 1 , are defined, respectively, as
H v ( a , b ) = a v b 1 v + a 1 v b v 2 , F α ( a , b ) = ( 1 α ) a b + α a + b 2 .

Note that H 0 ( a , b ) = H 1 ( a , b ) = a + b 2 (the arithmetic mean of a and b) and H 1 2 ( a , b ) = a b (the geometric mean of a and b). It is easy to see that as a function of v, H v ( a , b ) is convex, attains its minimum at v = 1 2 , and attains its maximum at v = 0 and v = 1 .

The operator version of the Heinz mean [1] asserts that if A, B and X are operators on a complex separable Hilbert space such that A and B are positive, then for every unitarily invariant norm , the function g ( v ) = A v X B 1 v + A 1 v X B v is convex on [ 0 , 1 ] , attains its minimum at v = 1 2 , and attains its maximum at v = 0 and v = 1 . Moreover, the operator version of the Heron mean [2] asserts that f ( α ) = ( 1 α ) A 1 2 X B 1 2 + α ( A X + X B 2 ) .

Let A , B , X M n , A, B are positive definite, Kaur and Singh [2] have proved the following inequalities for any unitarily invariant norm :
1 2 A v X B 1 v + A 1 v X B v ( 1 α ) A 1 2 X B 1 2 + α ( A X + X B 2 )
(1.1)
and
A 1 2 X B 1 2 1 2 A 2 3 X B 1 3 + A 1 3 X B 2 3 1 2 + t A X + t A 1 2 X B 1 2 + X B ,
(1.2)

where 1 4 v 3 4 , α [ 1 2 , ) and t ( 2 , 2 ] .

Replacing A, B by A 2 , B 2 in (1.1) and (1.2), then putting u = 2 v , the following inequalities hold:
1 2 A u X B 2 u + A 2 u X B u ( 1 α ) A X B + α ( A 2 X + X B 2 2 )
(1.3)
and
A X B 1 2 A 4 3 X B 2 3 + A 2 3 X B 4 3 1 t + 2 A 2 X + t A X B + X B 2 ,
(1.4)

where 1 2 u 3 2 , α [ 1 2 , ) and t ( 2 , 2 ] .

Zhan proved in [3] that if A , B , X M n , such that A, B are positive semidefinite, then
A u X B 2 u + A 2 u X B u 2 t + 2 A 2 X + t A X B + X B 2
(1.5)

for 1 2 u 3 2 and t ( 2 , 2 ] .

Let A , B , X M n , such that A, B are positive semidefinite, for 1 2 u 3 2 and t ( 2 , 2 ] ; Fu et al. in [4] proved that
2 A X B + 2 ( 1 2 3 2 A r X B 2 r + A 2 r X B r d r 2 A X B ) 2 t + 2 A 2 X + t A X B + X B 2 .
(1.6)

Recently, Kaur et al. [5], He et al. [6] and Bakherad et al. [7] have studied similar topics.

For the sake of convenience, we set
g ( u ) = A u X B 2 u + A 2 u X B u 2 .

In Section 2, we will generalize and refine some existing inequalities for unitarily invariant norms involving Heron and Heinz means for matrices and present some new refinements of the inequalities above.

2 Main results

In this section, we firstly utilize the convexity of the function g ( u ) to obtain a unitarily invariant norms inequality that leads to another version of the inequality (1.6), which is also the refinement of the inequality (1.5).

To obtain the results, we need the following lemma on convex functions [8, 9].

Lemma 2.1 Let f be a real valued continuous convex function on an interval [ a , b ] which contains ( x 1 , x 2 ) . Then for x 1 x x 2 , we have
f ( x ) f ( x 2 ) f ( x 1 ) x 2 x 1 x x 1 f ( x 2 ) x 2 f ( x 1 ) x 2 x 1 .
Theorem 2.2 Let A , B , X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 and α [ 1 2 , ) ,
A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ,
(2.1)

where r 0 = min [ u 2 , 1 u 2 ] .

Proof For 1 2 u 1 , by the convexity of the function g ( u ) and Lemma 2.1, presented above, we have
g ( u ) g ( 1 ) g ( 1 2 ) 1 2 u 1 2 g ( 1 ) g ( 1 2 ) ) 1 2 ,
which implies
g ( u ) 2 ( 1 u ) g ( 1 2 ) + ( 2 u 1 ) g ( 1 ) .
(2.2)
By (1.3) and (2.2), we have
A u X B 2 u + A 2 u X B u 4 ( 1 u ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 2 ( 2 u 1 ) A X B .
So,
A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
(2.3)
For 1 u 3 2 , by the convexity of the function g ( u ) and Lemma 2.1, presented above, we have
g ( u ) g ( 3 2 ) g ( 1 ) 1 2 u g ( 3 2 ) 3 2 g ( 1 ) 1 2 ,
which implies
g ( u ) ( 3 2 u ) g ( 1 ) + 2 ( u 1 ) g ( 3 2 ) .
(2.4)
By (1.3) and (2.4), we have
A u X B 2 u + A 2 u X B u 2 ( 3 2 u ) A X B + 4 ( u 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
So,
A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
(2.5)
By (2.3) and (2.5), for 1 2 u 3 2 , α [ 1 2 , ) and r 0 = min [ u 2 , 1 u 2 ] , we have the following equivalent inequality:
A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

The proof is completed. □

Remark 2.3 With a simple computation between the upper bounds in (1.3) and (2.1), obviously we have
( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ( 4 r 0 1 ) A X B 2 ( 1 2 r 0 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) = ( 4 r 0 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ( 4 r 0 1 ) A X B = ( 4 r 0 1 ) ( ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) A X B ) > 0 .

Thus the inequality (2.1) is a refinement of the inequality (1.3).

Now, we present a refinement of the inequality A X B ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

Theorem 2.4 Let A , B , X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 , and α [ 1 2 , ) , we have
2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) ,
(2.6)

where r 0 = min [ u 2 , 1 u 2 ] .

Proof For 1 2 u 1 , from Theorem 2.2, we have
A u X B 2 u + A 2 u X B u 4 ( 1 u ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 2 ( 2 u 1 ) A X B .
By integrating both sides of the inequality above, we have
1 2 1 A u X B 2 u + A 2 u X B u d u 4 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) 1 2 1 ( 1 u ) d u + 2 A X B 1 2 1 ( 2 u 1 ) d u = 1 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 1 2 A X B .
(2.7)
For 1 u 3 2 , from Theorem 2.2, we have
A u X B 2 u + A 2 u X B u 2 ( 3 2 u ) A X B + 4 ( u 1 ) ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .
Similarly, by integrating both sides of the inequality above, we have
1 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B 1 3 2 ( 3 2 u ) d u + 4 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) 1 3 2 ( 2 u 1 ) d u = 1 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + 1 2 A X B .
(2.8)
It follows from (2.7) and (2.8) that
1 2 3 2 A u X B 2 u + A 2 u X B u d u ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) + A X B ,
which is equivalent to
2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) 2 ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

The proof is completed. □

Remark 2.5 Obviously,
1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B 0 .

Thus, the inequality (2.6) is a refinement of the inequality A X B ( 1 α ) A X B + α ( A 2 X + X B 2 2 ) .

Taking α = 2 t + 2 ( 2 < t 2 ), the following corollaries are obtained.

Corollary 2.6 Let A , B , X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm and 1 2 u 3 2 ,
A u X B 2 u + A 2 u X B u 2 ( 4 r 0 1 ) A X B + 4 ( 1 2 r 0 ) t + 2 A 2 X + t A X B + X B 2 ,
(2.9)

where r 0 = min [ u 2 , 1 u 2 ] and 2 < t 2 .

Corollary 2.7 Let A , B , X M n , such that A, B are positive semidefinite. Then for any unitarily invariant norm , 1 2 u 3 2 ,
2 A X B + 2 ( 1 2 3 2 A u X B 2 u + A 2 u X B u d u 2 A X B ) 2 t + 2 A 2 X + t A X B + X B 2 ,
(2.10)

where r 0 = min [ u 2 , 1 u 2 ] and 2 < t 2 .

Thus, on the one hand, the inequality (2.9) is a refinement of the inequality A X B 1 t + 2 A 2 X + t A X B + X B 2 , and also another version of the inequality (1.6); on the other hand, the inequality (2.10) is just the inequality proved in [4], so the inequality (2.6) presented in Theorem 2.4 is also the generalization of the inequality proved in [4].

Declarations

Acknowledgements

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing University

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© Cao and Wu; licensee Springer 2014

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